Algebraicity of harmonic Maass forms

In 1947 D.H. Lehmer conjectured that Ramanujan’s tau-function never vanishes. In the 1980s, B. Gross and D. Zagier proved a deep formula expressing the central derivative of suitable Hasse–Weil L-functions in terms of the Neron–Tate height of a Heegner point. This expository article describes recent work (with J.H. Bruinier and R. Rhoades) which reformulates both topics in terms of the algebraicity of harmonic Maass forms.     

Eichler integrals and harmonic weak Maass forms

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, H-groups by employing the theory of supplementary functions introduced and developed by M.I. Knopp and S.Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 ? k. The operator ξ_2-k (resp. D ^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ_2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D ^k-1.     

Deformations of Maass forms

We describe numerical calculations which examine the Phillips-Sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume Riemann surface under deformation of the surface. Our calculations indicate that if the Teichmüller space of is not trivial, then each cusp form has a set of deformations under which either the cusp form remains a cusp form or else it dissolves into a resonance whose constant term is uniformly a factor of smaller than a typical Fourier coefficient of the form. We give explicit examples of those deformations in several cases.

     

Harmonic Maass forms and periods

According to Waldspurger’s theorem, the coefficients of half-integral weight eigenforms are given by central critical values of twisted Hecke $L$ -functions, and therefore by periods. Here we prove that the coefficients of the holomorphic parts of weight $1/2$ harmonic Maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves.     

Exponential sums involving Maass forms

We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.     

Pfister forms and their applications

This note investigates the properties of the 2ⁿ-dimensional quadratic forms ?_i=2ⁿ 〈1, a_i〉, called n-fold Pfister forms. Utilizing these properties, various applications are made to k-theory of fields, and field invariants. In particular, the set of orderings of a field and the maximum dimension of anisotropic forms with everywhere zero signature are investigated. Details will appear elsewhere.     

Heegner points,cycles and Maass forms

We derive for Hecke-Maass cusp forms on the full modular group a relation between the sum of the form at Heegner points (and integrals over Heegner cycles) and the product of two Fourier coefficients of a corresponding form of half-integral weight. Specializing to certain cycles we obtain the nonnegativity of theL-function of such a form at the center of the critical strip. These results generalize similar formulae known for holomorphic forms. Partially supported by NSF grant # DMS-9096262. Partially supported by NSF grant # DMS-9102082.     

Identities of cycle integrals of weak Maass forms

The Ramanujan Journal - We prove identities between cycle integrals of non-holomorphic modular forms arising from applications of various differential operators to weak Maass forms     

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of ∑m12+m22+m32≤xλ(m12+m22+m32)Λ(m12+m22+m32) and ∑m12+m22+m32≤xa(m12+m22+m32)Λ(m12+m22+m32).     

Zero density of L-functions related to Maass forms

Let f(z) be a Hecke-Maass cusp form for SL ₂(?), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ? T, β ? σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ? σ ? 1 ? ?, there exists a constant C = C(?) such that N(σ,T) ? T ^2(1?σ)/σ(logT) ^C , which improves the previous results.     

On the effective determination of Maass cusp forms by L-values

Let f be a Hecke–Maass cusp form of Laplace eigenvalue 1/4+μ ² with |μ|≤Λ for \(\mathit{SL}_{2}(\mathbb{Z})\) . We show that f is uniquely determined by the central values of Rankin–Selberg L-functions L(s,f?g), where g runs over the set of holomorphic cusp forms of weight k? _? Λ ^1+3θ+? for any ?>0 for \(\mathit{SL}_{2}(\mathbb{Z})\) .     

Twisted traces of CM values of weak Maass forms

We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by Zagier on traces of singular moduli. We utilize a twisted version of the theta lift considered by Bruinier and Funke (2006) [BF06].